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Quadratic Inequalities

Graphs and sign tests for squared functions greater or less than a number.

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Compete the table.
Word Definition
Quadratic Inequality ___________________________________________________
___________ the values of that make equal to zero in a quadratic function
Polynomial Inequality: ___________________________________________________
___________ A function which may be expressed as a ratio of two polynomials, specified to be greater or less than a given value

Quadratic Inequalities

To solve quadratic inequalitites, you can graph it and see visually where the inequality is true. Without graphing, you can follow these steps:

  1. Set up the inequality in the form    (or  \begin{align*}p(x)<0, p(x)\le0,p(x)\ge0\end{align*} )
  2. Find the solutions to the equation \begin{align*}p(x)=0\end{align*} .
  3. Divide the number line into intervals based on the solutions to \begin{align*}p(x)=0\end{align*} .
  4. Use test points to find solution sets to the equation.


 Find the solution set for the equation .

First, factor:             

\begin{align*}x^2 + 2x - 8 = 0\end{align*}

\begin{align*}(x + 4)(x - 2) = 0\end{align*}

Then create the intervals: 

 \begin{align*}(-\infty,-4) | (-4,2) | (2,\infty)\end{align*}

Finally, make a chart.

Interval Test Point Is \begin{align*}x^{2}+2x-8\end{align*} positive or negative? Part of Solution set?
\begin{align*}(-\infty,-4)\end{align*} \begin{align*}-5\end{align*} \begin{align*}+\end{align*} \begin{align*}yes\end{align*}
\begin{align*}(-4, 2)\end{align*} \begin{align*}1\end{align*} \begin{align*}-\end{align*} \begin{align*}no\end{align*}
\begin{align*}(2,+\infty)\end{align*} \begin{align*}3\end{align*} \begin{align*}+\end{align*} \begin{align*}yes\end{align*}

  if and only if \begin{align*}x < -4\end{align*} and  .


For more on quadratic inequalities, click here

Polynomial and Rational Inequalities

In polynomial and rational inequalities, the four basic steps are the same as in polynomial inequalities. For rational inequalities, however, you must also account for possible sign changes at vertical asymptotes or a break in the graph.

Add the vertical asymtotes to your intervals, and create the same chart as you did for quadratic inequalities. 


For more on polynomial and rational inequalities, click here


1) Find the solution set of the inequality \begin{align*}x^2 \leq 36\end{align*}

2) Find the solution set: 

3) Find the solution set: 

4) Graph the solution set: \begin{align*}(x - 3)(x + 4) \geq 0\end{align*}


Find the solution set of the following inequalities without using a calculator. Display the solution set on the number line.
  1. \begin{align*}x^{2}+2x-3\le0\end{align*}
  2. \begin{align*}-6x^{2}-13x+5\ge0\end{align*}
  3. \begin{align*}\frac{1-x}{x}<1\end{align*}
  4.  \begin{align*}\frac{x^4}{4} - x^2 <0\end{align*}
  5. \begin{align*}4x^3 - 8x^2 - x + 2 \geq 0 \end{align*}
Solve the following inequalities:
  1. \begin{align*}\frac{n^3 - 2n^2 - n + 2}{n^3 + 3n^2 + 4n + 12} < 0 \end{align*}
  2. \begin{align*}\frac{n^3 + 3n^2 - 4n - 12}{n^3 - 5n^2 + 4n - 20} \leq 0 \end{align*}
  3. \begin{align*}\frac{2n^3 + 5n^2 - 18n - 45}{3n^3 - n^2 +27n - 9} \geq 0 \end{align*}

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